Problem 16 - Applied Calculus

Question

Solve each equation by factoring. [Hint for: First factor out a fractional power.] $$ 3 x^{4}+12 x^{2}=12 x^{3} $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 2\).

Step 1: Rearrange the Equation

Start by bringing all terms to one side of the equation so that the equation is set to zero: \[3x^4 + 12x^2 - 12x^3 = 0\]

Step 2: Factor Out the Greatest Common Factor

Identify the greatest common factor (GCF) in all terms. Here, the GCF is \(3x^2\). Factoring this out gives:\[3x^2(x^2 - 4x + 4) = 0\]

Step 3: Factor the Quadratic Expression Completely

The expression inside the parentheses, \(x^2 - 4x + 4\), is a perfect square trinomial. It can be factored as:\[(x - 2)^2\]So the equation becomes:\[3x^2(x - 2)^2 = 0\]

Step 4: Solve Each Factor Set to Zero

To find the solution, set each factor equal to zero:\[3x^2 = 0\]\[x - 2 = 0\]Solving these gives:\[x = 0\] (from \(3x^2 = 0\))\[x = 2\] (from \(x - 2 = 0\))

Step 5: Conclude the Solution

Considering all factors equal to zero, the solutions are \(x = 0\) with multiplicity 1 and \(x = 2\) with multiplicity 2. Thus, the set of solutions is \(x = 0\) and \(x = 2\).

Key Concepts

Quadratic equations

Quadratic equations are fundamental in algebra and appear as terms in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a\) not equal to zero. These equations are quadratic because the highest exponent of the variable \(x\) is two.
In tackling quadratic equations, factoring is a popular method to find their solutions. To factor a quadratic equation, you express it as a product of two binomials. For example, \(x^2 - 4x + 4\) can be factorized to \((x - 2)(x - 2)\).
The solutions, or roots, of the quadratic equation are the values of \(x\) that make the equation true when substituted back in. You often achieve these by setting each factor equal to zero, leading you to the solutions of the equation.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is one of the first steps in solving polynomial equations by factoring. The GCF is the highest expression that divides each term of the polynomial without leaving any remainder. It's like simplifying the equation by taking out the commonality across all terms.
For the equation \(3x^4 + 12x^2 - 12x^3\), the GCF is \(3x^2\). Breaking down each term confirms this:

• For \(3x^4\), the factors are 3, \(x\), \(x\), \(x\), \(x\).
• For \(12x^2\), the factors are 3, 2, 2, \(x\), \(x\).
• For \(12x^3\), the factors are 3, 2, 2, \(x\), \(x\), \(x\).

By factoring out \(3x^2\), the equation simplifies to \(3x^2(x^2 - 4x + 4) = 0\), making it easier to proceed finding further factors or solving the equation.

Perfect square trinomial

A perfect square trinomial is a special kind of quadratic expression. It can be written as the square of a binomial. Recognizing these helps simplify factoring and solving equations.
A perfect square trinomial has the form \(a^2 - 2ab + b^2 = (a - b)^2\) or \(a^2 + 2ab + b^2 = (a + b)^2\). Identifying the pattern is key. For example, in the expression \(x^2 - 4x + 4\), we recognize it as a perfect square because:

• The first term \(x^2\) is a perfect square of \(x\).
• The last term 4 is a perfect square of 2.
• The middle term -4x is twice the product of \(x\) and 2.

Thus, this trinomial becomes \((x - 2)^2\). Understanding perfect square trinomials not only aids in factoring but also speeds up solving quadratic equations.